1 Answer
1

The purpose of the reconstruction of the polynomial $P(x)$, is just to calculate the value of $P(0)$, which equals the shared secret value.

If Lagrange polynomials are used, a trivial optimization which cuts the number of multiplication nearly in half is
$$P(0) = (\prod_{i=0}^{n-1}{-x_i})\sum_{i=0}^{n-1}{{\frac{y_i}{-x_i (\prod_{j=0,j\neq i}^{n-1}{(x_i-x_j)})}}}.$$

If Newton polynomials are used, there is no trivial optimization because you only want to calculate $P(0)$, simply because the formula is already optimized to half of $n^2$. This means that the overhead of the recursion might outweigh the general benefits compare to the Lagrange formula.

If you had had to represent the polynomial $P(x)$ in order to calculate other values than just $P(0)$, Newton polynomials would in general have been more efficient.

Thanks, I improved it a bit further, to make it a bit a clearer exactly which calculations are made.
–
Henrick HellströmJun 25 '14 at 11:17

1

Technically, your question captures that $O(n^2) = O(kn^2 + r)$ for constant parameters $k, r$. The formula in my answer requires only about half as many multiplications, as if you were to calculate the coefficients of $P(x)$ instead of the the value of $P(0)$ directly. Newton's formula calculates the coefficients faster than Lagrange's formula, but it doesn't necessarily calculate the value of $P(0)$ faster.
–
Henrick HellströmJun 25 '14 at 12:35

@HenrickHellström Sorry, I deleted my comment a few minutes after posting, as I managed to figure it out! Thanks though!
–
KateJun 25 '14 at 12:37